Spin-split Landau Levels Research Articles

Magnetic field dependent two-photon absorption properties in monolayer MoS2

We report our theoretical investigations on the magnetic field dependence of two-photon absorption (TPA) properties in ${\mathrm{MoS}}_{2}$. Based on the Landau levels obtained by the effective massive Dirac model, we developed TPA theories associated with both interband transitions and intraband transitions for transitional metal dichalcogenides (TMDCs) with a magnetic field under the second-order perturbation theory with respect to the electron-photon interaction. Our results reveal that the TPA coefficient is effectively enhanced by five orders of magnitude under the application of an external magnetic field, which results in the quantized horizontal Landau levels with the spin-orbit coupling (SOC) and valley Zeeman effects. The TPA peaks are located in the terahertz regime for intraband transitions and in the visible regime for interband transitions. The amplitude of TPA contributed by intraband transitions is much larger than that of interband transitions owing to the resonant transitions. Oscillatory behavior appeared in the interband transition-related TPA spectra. The separation of TPA for the transitions in the spin-up and spin-down states caused by the spin splitting of Landau levels resulting from the strong SOC is illustrated. We also analyzed the anisotropy of TPA associated with the direction of the incident light. Our results are of much importance for the application of TMDCs in optical devices.

Spin-valley locking and bulk quantum Hall effect in a noncentrosymmetric Dirac semimetal BaMnSb2

Spin-valley locking in monolayer transition metal dichalcogenides has attracted enormous interest, since it offers potential for valleytronic and optoelectronic applications. Such an exotic electronic state has sparsely been seen in bulk materials. Here, we report spin-valley locking in a Dirac semimetal BaMnSb2. This is revealed by comprehensive studies using first principles calculations, tight-binding and effective model analyses, angle-resolved photoemission spectroscopy measurements. Moreover, this material also exhibits a stacked quantum Hall effect (QHE). The spin-valley degeneracy extracted from the QHE is close to 2. This result, together with the Landau level spin splitting, further confirms the spin-valley locking picture. In the extreme quantum limit, we also observed a plateau in the z-axis resistance, suggestive of a two-dimensional chiral surface state present in the quantum Hall state. These findings establish BaMnSb2 as a rare platform for exploring coupled spin and valley physics in bulk single crystals and accessing 3D interacting topological states.

Spin relaxation of holes in In0.53Ga0.47As/InP quantum wells

Polarization-resolved photoluminescence was used to study spin relaxation of photoexcited holes in In0.53Ga0.47As/InP quantum wells in a quantizing magnetic field as a function of temperature. At a temperature below 10 K, the circular polarization of the photoluminescence due to the spin-split valence band Landau levels was found temperature-independent. In this temperature range fast hole spin relaxation as compared to their lifetime leads to the photoluminescence circular polarization determined by the ratio of these times. Increasing temperature resulted in efficient hole spin thermalization in the Zeeman split valence band Landau levels and as a consequence, in vanishing photoluminescence polarization. Fits of the experimental data by the theory allowed a determination of the hole spin relaxation times related to different Landau levels and the corresponding hole effective g-factor. Direct measurements of the hole spin relaxation times prove the obtained results.

Size scaling of the exchange interaction in the quantum Hall effect regime

We use the self-consistent Hartree-Fock approximation for numerically addressing the integer quantum Hall effect (IQHE) regime in terms of many-body physics at higher Landau levels (LLs). We investigate the dependence of many-particle interactions on the lateral size of the electron system. We use the exchange enhancement of the $g$-factor for spin-polarized Landau levels as an indicator for the strength of the exchange interaction. The driving force for the $g$-factor enhancement is a Hund's rule behavior for the occupation of spin-split Landau levels that lowers the many-particle ground state energy by arranging as many spins in parallel as possible. By increasing the total number of electrons and total number of available states per LL, it can therefore be expected that the exchange-enhanced spin gap should increase as well. In contrast to the dependence on the magnetic field, an increase of the total number of states by simply increasing the system size at constant magnetic field shows a clear saturation behavior above a lateral system size of 1000 nm. The importance of this result is underlined by an extended introduction, which demonstrates the permanent dominance of many-body interactions in all transport regimes of the IQHE. A modeling of IQHE systems therefore has to include many-body interactions, and our results open a pathway towards many-body modeling of quantum Hall systems of macroscopic size.

Nonperturbative Matrix Mechanics Approach to Spin-Split Landau Levels and the g Factor in Spin-Orbit Coupled Solids.

We propose a fully quantum approach to nonperturbatively calculate the spin-split Landau levels and g factor of various spin-orbit coupled solids based on the k·p theory in the matrix mechanics representation. The new method considers the detailed band structure and the multiband effect of spin-orbit coupling irrespective of the magnetic-field strength. We show an application of this method to PbTe, a typical Dirac electron system. Contrary to popular belief, we show that the spin-splitting parameter M, which is the ratio of the Zeeman to cyclotron energy, exhibits a remarkable magnetic-field dependence. This field dependence can rectify the existing discrepancy between experimental and theoretical results. We also show that M evaluated from the fan diagram plot is different from that determined as the ratio of the Zeeman to cyclotron energy, which also overturns common belief.

Impact of antiferromagnetic order on Landau-level splitting of quasi-two-dimensional Dirac fermions in EuMnBi2

We report spin-split Landau levels of quasi-two-dimensional Dirac fermions in a layered antiferromagnet EuMnBi$_2$, as revealed by interlayer resistivity measurements in a tilted magnetic field up to $\sim$35 T. The amplitude of Shubnikov-de Haas (SdH) oscillation in interlayer resistivity is strongly modulated by changing the tilt angle of the field, i.e., the Zeeman-to-cyclotron energy ratio. The effective $g$ factor estimated from the tilt angle, where the SdH oscillation exhibits a phase inversion, differs by approximately 50% between two antiferromagnetic phases. This observation signifies a marked impact of the magnetic order of Eu sublattice on the Dirac-like band structure. The origin may be sought in strong exchange coupling with the local Eu moments, as verified by the first-principles calculation.

Quantum electron transport in magnetically entangled subbands

Transport properties of highly mobile two-dimensional (2D) electrons in symmetric GaAs quantum wells with two populated subbands placed in tilted magnetic fields are studied at high temperatures. Quantum positive magnetoresistance (QPMR) and magneto-intersubband resistance oscillations (MISO) are observed in quantizing magnetic fields, ${B}_{\ensuremath{\perp}}$, applied perpendicular to the 2D layer. QPMR displays contributions from electrons with considerably different quantum lifetimes, ${\ensuremath{\tau}}_{q}^{(1,2)}$, confirming the presence of two subbands in the studied system. MISO evolution with ${B}_{\ensuremath{\perp}}$ agrees with the obtained quantum scattering times only if an additional reduction of the MISO magnitude is applied at small magnetic fields. This indicates the presence of a yet unknown mechanism leading to MISO damping. Application of an in-plane magnetic field produces a strong decrease of both QPMR and MISO magnitude. The reduction of QPMR is explained by spin splitting of Landau levels indicating a $g$ factor, $g\ensuremath{\approx}0.4$, which is considerably less than the $g$ factor found in GaAs quantum well with a single subband populated. In contrast to QPMR, the decrease of MISO magnitude is largely related to the in-plane magnetic field induced entanglement between quantum levels in different subbands that, in addition, increases the MISO period.

Quantum Hall effect in black phosphorus two-dimensional electron system.

The development of new, high-quality functional materials has been at the forefront of condensed-matter research. The recent advent of two-dimensional black phosphorus has greatly enriched the materials base of two-dimensional electron systems (2DESs). Here, we report the observation of the integer quantum Hall effect in a high-quality black phosphorus 2DES. The high quality is achieved by embedding the black phosphorus 2DES in a van der Waals heterostructure close to a graphite back gate; the graphite gate screens the impurity potential in the 2DES and brings the carrier Hall mobility up to 6,000 cm(2) V(-1) s(-1). The exceptional mobility enabled us to observe the quantum Hall effect and to gain important information on the energetics of the spin-split Landau levels in black phosphorus. Our results set the stage for further study on quantum transport and device application in the ultrahigh mobility regime.

Landau level transitions in InAs/AlSb/GaSb quantum wells**Project supported by the National Natural Science Foundation of China (Grant Nos. 61076092 and 61290303).

The electronic structure of InAs/AlSb/GaSb quantum wells embedded in AlSb barriers and in the presence of a perpendicular magnetic field is studied theoretically within the 14-band k·p approach without making the axial approximation. At zero magnetic field, for a quantum well with a wide InAs layer and a wide GaSb layer, the energy of an electron-like subband can be lower than the energy of hole-like subbands. As the strength of the magnetic field increases, the Landau levels of this electron-like subband grow in energy and intersect the Landau levels of the hole-like subbands. The electron–hole hybridization leads to a series of anti-crossing splittings of the Landau levels. The magnetic field dependence of some dominant transitions is shown with their corresponding initial-states and final-states indicated. The dominant transitions at high fields can be roughly viewed as two spin-split Landau level transitions with many electron–hole hybridization-induced splittings. When the magnetic field is tilted, the electron-like Landau level transitions show additional anti-crossing splittings due to the subband-Landau level coupling.

Magneto-Optics of Massive Dirac Fermions in BulkBi2Se3

We report on magneto-optical studies of Bi2Se3, a representative member of the 3D topological insulator family. Its electronic states in bulk are shown to be well described by a simple Dirac-type Hamiltonian for massive particles with only two parameters: the fundamental band gap and the band velocity. In a magnetic field, this model implies a unique property-spin splitting equal to twice the cyclotron energy: Es=2Ec. This explains the extensive magnetotransport studies concluding a fortuitous degeneracy of the spin and orbital split Landau levels in this material. The Es=2Ec match differentiates the massive Dirac electrons in bulk Bi2Se3 from those in quantum electrodynamics, for which Es=Ec always holds.

Exchange enhancement of the electron g factor in strained InGaAs/InP heterostructures

The exchange enhancement of the electron g factor in strained InGaAs/InP heterostructures with a two-dimensional electron gas is studied. Analysis of the temperature dependence of the resistance in the minima of the Shubnikov-de Haas oscillations in perpendicular magnetic fields up to 12 T in the vicinity of the odd filling factors of the Landau levels yields the values of the effective electron Lande factor g* from −8.6 to −10.1. The experimental values are compared with the results of theoretical calculations of the g factor of quasiparticles. The calculations are performed using an eight-band k · p Hamiltonian and take into account exchange interaction in the two-dimensional electron gas. It is shown that, under the conditions of a large overlap between the spin-split Landau levels, the maximum value of the quasiparticle g factor can be attained in the vicinity of even filling factors. This is caused by the nonparabolicity of the electron dispersion relation.

Observation of Rashba zero-field spin splitting in a strained germanium 2D hole gas

We report the observation, through Shubnikov-de Haas oscillations in the magnetoresistance, of spin splitting caused by the Rashba spin-orbit interaction in a strained Ge quantum well epitaxially grown on a standard Si(001) substrate. The Shubnikov-de Haas oscillations display a beating pattern due to the spin split Landau levels. The spin-orbit parameter and Rashba spin-splitting energy are found to be 1.0 × 10−28 eVm3 and 1.4 meV, respectively. This energy is comparable to 2D electron gases in III-V semiconductors, but substantially larger than in Si, and illustrates the suitability of Ge for modulated hole spin transport devices.

Observation of three-dimensional massless Kane fermions in a zinc-blende crystal

Solid state physics and quantum electrodynamics with its ultra-relativistic (massless) particles meet, to their mutual beneit, in the electronic properties of one-dimensional carbon nanotubes as well as two-dimensional graphene or surfaces of topological insulators. However, clear experimental evidence for electronic states with conical dispersion relations in all three dimensions, conceivable in certain bulk materials, is still missing. In the present work, we fabricate and study a zinc-blend crystal, HgCdTe, at the point of the semiconductor-to-semimetal topological transition. Three-dimensional massless electrons with a velocity of about 10$^6$ m/s are observed in this material, as testifed by: (i) the dynamical conductivity which increases linearly with the photon frequency, (ii) in a magnetic field $B$, by a $\sqrt{B}$ dependence of dipole-active inter-Landau-level resonances and (iii) the spin splitting of Landau levels, which follows a $\sqrt{B}$ dependence, typical of ultra-relativistic particles but not really seen in any other electronic system so far.

Analysis of the spin splitting of maxima of quantum oscillations of the resistivity of n-Bi-Sb semiconductor alloys in a magnetic field parallel to the bisector axis

In samples of semiconductor alloys n-Bi0.93Sb0.07 with different electron concentrations (n1 = 8 × 1015 cm−3, n2 = 1.2 × 1017 cm−3, and n3 = 1.9 × 1018 cm−3), dependences of the electrical resistivity on magnetic fields up to 45 T parallel to the current and the bisector axis (H ‖ C1 ‖ j) have been measured at temperatures of 1.5, 4.5, and 10 K. The obtained dependences ρ22(H) demonstrate quantum oscillations of the resistivity (Shubnikov-de Haas effect), and, in high magnetic fields, there is a resistivity maximum far away from other maxima. On assumption that this maximum is related to the spin-split Landau level N = 0− for electrons of the main ellipsoid, the spin-splitting parameters are calculated for electrons of the main ellipsoid: γ1 = 0.87, γ2 = 0.8, and γ3 = 0.73. Using these values, the oscillation maxima can be reliably related to the numbers of split Landau levels for electrons of the main and secondary ellipsoids. The dependences of the resistivity ρ11 and the Hall coefficient R31.2 on magnetic field have been measured in a transverse magnetic field at H ‖ C1 and j ‖ C2 on the sample with the electron concentration n4 = 1.4 × 1017 cm−3. Using similar analysis, the spin-splitting parameter is found to be γ4 = 0.85, which is close to the value of γ2 = 0.8 obtained for the sample with close electron concentration (n2 = 1.2 × 1017 cm−3) during the measurements in a longitudinal magnetic field. The quantum oscillation maxima of Hall coefficient R31.2 are shifted to the range of high magnetic fields as compared to the quantum oscillation maxima of resistivity ρ11.

Direct measurement of the spin gaps in a gated GaAs two-dimensional electron gas

We have performed magnetotransport measurements on gated GaAs two-dimensional electron gases in which electrons are confined in a layer of the nanoscale. From the slopes of a pair of spin-split Landau levels (LLs) in the energy-magnetic field plane, we can perform direct measurements of the spin gap for different LLs. The measured g-factor g is greatly enhanced over its bulk value in GaAs (0.44) due to electron–electron (e-e) interactions. Our results suggest that both the spin gap and g determined from conventional activation energy studies can be very different from those obtained by direct measurements.

Broadening and splitting of emission spectra of a GaInAs/AlInAs quantum cascade laser in a quantising magnetic field

We have studied the effect of a relatively weak quantising magnetic field on emission spectra of a GaInAs/AlInAs quantum cascade laser near 10 K. The results demonstrate that, as the magnetic field induction increases to 7 T, the spectra broaden (to 5 meV) and split into three emission bands. As a result, we observe simultaneously up to 80 longitudinal lasing modes in the three bands and the integrated laser output intensity increases 70 times. The presence of bands in the emission spectra can be accounted for in terms of the magnetic quantisation of the laser levels into spinsplit Landau levels. The increase in emission intensity is attributable primarily to phonon resonance adjustment in a magnetic field.

Long spin relaxation time of holes in InGaAs/GaAs quantum wells probed by cyclotron resonance spectroscopy

In this paper, we report on a long, ms range, hole spin relaxation time in InGaAs/GaAs quantum wells probed by cyclotron resonance spectroscopy in pulsed magnetic fields. In our experiments, we found strong hysteresis in the spectral weights of cyclotron resonance absorption lines when rapidly changing magnetic field is used for the experiment. The hysteresis vanishes when a much slower changing magnetic field is used. We attribute this behavior to a long energy relaxation time between the two lowest spin-split hole Landau levels, i.e., a long hole spin relaxation time. We also present transition frequencies calculated using a $4\ifmmode\times\else\texttimes\fi{}4$ Luttinger Hamiltonian, which confirm our findings.

Chiral condensate with topological degeneracy in graphene and its manifestation in edge states

The role of chiral symmetry in many-body states of graphene in strong magnetic fields is theoretically studied with the honeycomb lattice model. For a spin-split Landau level where the leading electron-electron interaction is the nearest-neighbor repulsion, a chiral condensate is shown to be, within the subspace of the $n=0$ Landau level, an exact many-body ground state having a finite gap, for which a calculation of the Chern number reveals that the ground state is a Hall insulator with a topological degeneracy of two. The topological nature of the ground state is shown to manifest itself as a Kekul\'ean bond order along armchair edges, while the pattern melts in the bulk due to quantum fluctuations. These can be regarded as a realization of the bulk-edge correspondence that is peculiar to a chiral-symmetric system. We have also obtained the ground state when point defects are introduced in the honeycomb lattice to reveal how the presence or absence of the chiral symmetry affects the defect states.

The effect of exchange interaction on quasiparticle Landau levels in narrow-gap quantum well heterostructures

Using the ‘screened’ Hartree–Fock approximation based on the eight-band k⋅p Hamiltonian, we have extended our previous work (Krishtopenko et al 2011 J. Phys.: Condens. Matter 23 385601) on exchange enhancement of the g-factor in narrow-gap quantum well heterostructures by calculating the exchange renormalization of quasiparticle energies, the density of states at the Fermi level and the quasiparticle g-factor for different Landau levels overlapping. We demonstrate that exchange interaction yields more pronounced Zeeman splitting of the density of states at the Fermi level and leads to the appearance of peak-shaped features in the dependence of the Landau level energies on the magnetic field at integer filling factors. We also find that the quasiparticle g-factor does not reach the maximum value at odd filling factors in the presence of large overlapping of spin-split Landau levels. We advance an argument that the behavior of the quasiparticle g-factor in weak magnetic fields is defined by a random potential of impurities in narrow-gap heterostructures.

The influence of the Rashba spin–orbit coupling on the two-dimensional magnetoexcitons

The influence of the Rashba spin–orbit coupling (RSOC) on the two-dimensional (2D) electrons and holes in a strong perpendicular magnetic field leads to different results for the Landau quantization in different spin projections. In the Landau gauge the unidimensional wave vector describing the free motion in one in-plane direction is the same for both spin projections, whereas the numbers of Landau quantization levels are different. For an electron in an s-type conduction band they differ by one, as was established earlier by Rashba (1960 Fiz. Tverd. Tela 2 1224), whereas for heavy holes in a p-type valence band influenced by the 2D symmetry of the layer they differ by three. The shifts and the rearrangements of the 2D hole Landau quantization levels on the energy scale are much larger in comparison with the case of conduction electron Landau levels. This is due to the strong influence of the magnetic field on the RSOC parameter. At sufficiently large values of this parameter the shifts and rearrangements are comparable with the hole cyclotron energy. There are two lowest spin-split Landau levels for electrons as well as four lowest ones for holes in the case of small RSOC parameters. They give rise to eight lowest energy bands of the 2D magnetoexcitons, as well as of the band-to-band quantum transitions. It is shown that three of them are dipole-active, three are quadrupole-active and two are forbidden. The optical orientation under the influence of circularly polarized light leads to optical alignment of the magnetoexcitons with different orbital momentum projections in the direction of the external magnetic field.